On Noether and strict stability, Hilbert exponent, and relative Nullstellensatz

Chia-chi Tung¹

¹ Department of Mathematics and Statistics Minnesota State University, Mankato Mankato, MN 56001, U.S.A.

Annales Polonici Mathematici, Tome 107 (2013) no. 1, pp. 1-28.

Résumé

Conditions characterizing the membership of the ideal of a subvariety ${\mathfrak S}$ arising from (effective) divisors in a product complex space $Y \times X$ are given. For the algebra ${\mathcal O}_Y [V]$ of relative regular functions on an algebraic variety $V$, the strict stability is proved, in the case where $Y$ is a normal space, and the Noether stability is established under a weakened condition. As a consequence (for both general and complete intersections) a global Nullstellensatz is derived for divisors in $Y\times {\mathbb C}^N$, respectively, $Y\times {\mathbb P}^N ({\mathbb C})$. Also obtained are a principal ideal theorem for relative divisors, a generalization of the Gauss decomposition rule, and characterizations of solid pseudospherical harmonics on a semi-Riemann domain. A result towards a more general case is as follows: Let ${\mathfrak D}_j$, $1 \le j\le p$, be principal divisors in $X$ associated to the components of a $q$-weakly normal map $g = (g_1,\ldots,g_p) : X \to {\mathbb C}^p$, and $S := \bigcap {\mathfrak S}_{|{\mathfrak D}_j|}$. Then for any proper slicing $(\varphi,g,D)$ of $\{{\mathfrak D}_j\}_{1\le j\le p}$ (where $D\subset X$ is a relatively compact open subset), there exists an explicitly determined Hilbert exponent ${\mathfrak h}_{_{{\mathfrak D}_1 \cdots {\mathfrak D}_p,D}}$ for the ideal of the subvariety ${\mathfrak S} = Y\times (S\cap D)$.

Détail

DOI : 10.4064/ap107-1-1

Keywords: conditions characterizing membership ideal subvariety mathfrak arising effective divisors product complex space times given algebra mathcal relative regular functions algebraic variety strict stability proved where normal space noether stability established under weakened condition consequence general complete intersections global nullstellensatz derived divisors times mathbb respectively times mathbb mathbb obtained principal ideal theorem relative divisors generalization gauss decomposition rule characterizations solid pseudospherical harmonics semi riemann domain result towards general follows mathfrak principal divisors associated components q weakly normal map ldots mathbb bigcap mathfrak mathfrak proper slicing varphi mathfrak where subset relatively compact subset there exists explicitly determined hilbert exponent mathfrak mathfrak cdots mathfrak ideal subvariety mathfrak times cap